Let
L
\mathcal {L}
be a complete projective logic. Then
L
\mathcal {L}
has a natural representation as the lattice of
⟨
⋅
,
⋅
⟩
\langle { \cdot , \cdot } \rangle
-closed subspaces of a left vector space V over a division ring D, where
⟨
⋅
,
⋅
⟩
\langle {\cdot ,\cdot } \rangle
is a definite
θ
\theta
-bilinear symmetric form on V,
θ
\theta
being some involutive antiautomorphism of D. Now a well-known theorem of Piron states that if D is isomorphic to the real field, the complex field or the sfield of quaternions, if
θ
\theta
is continuous, and if the dimension of
L
\mathcal {L}
is properly restricted, then
L
\mathcal {L}
is just one of the standard Hilbert space logics. Here we also assume
L
\mathcal {L}
is a complete projective logic. Then if every
θ
\theta
-fixed element of D is in the center of D and can be written as
±
d
θ
(
d
)
\pm \,d\theta (d)
, some
d
∈
D
d \in D
, and if the dimension of
L
\mathcal {L}
is properly restricted, we show that
L
\mathcal {L}
is just one of the standard Hilbert space logics over the reals, the complexes, or the quaternions. One consequence is the extension of Piron’s theorem to discontinuous
θ
\theta
. Another is a purely lattice theoretic characterization of the lattice of closed subspaces of separable complex Hilbert space.